Improved constructions of mixed state quantum automata

Quantum finite automata with mixed states are proved to be super-exponentially more concise rather than quantum finite automata with pure states. It was proved earlier by A. Ambainis and R. Freivalds that quantum finite automata with pure states can have an exponentially smaller number of states than deterministic finite automata recognizing the same language. There was an unpublished “folk theorem” proving that quantum finite automata with mixed states are no more super-exponentially more concise than deterministic finite automata. It was not known whether the super-exponential advantage of quantum automata is really achievable.We prove that there is an infinite sequence of distinct integers n, languages Ln, and quantum finite automata with mixed states with 5n states recognizing language Ln with probability , while any deterministic finite automaton recognizing Ln needs at least states.Unfortunately, the alphabet for these languages grows with n. In order to prove a similar result for languages in a fixed alphabet we consider a counterpart of Hamming codes for permutations of finite sets, i.e. sets of permutations such that any two distinct permutations in the set have Hamming distance at least d. The difficulty arises from the fact that in the traditional Hamming codes for binary strings, positions in the string are independent while positions in a permutation are not independent. For instance, any two permutations of the same set either coincide or their Hamming distance is at least 2. The main combinatorial problem still remains open.